Character tables

how does one go about constructing the character table for a group? while only knowing the group, and the classes. I was thinking of starting to construct representations and diagonalizing them but that seems like it would take an exorbitant amount of time.

is there a method for constructing the character table without knowing the irreducible representations?

sorry if this should be moved, this is related to a homework problem, but as its talking more about methods I thought it could go here.

Also on asimilar note if the group contains an invariant subgroup how does this affect the character table

Edit: I realised this is a physicist thing, but the character is the trace of a matrix representation, and the class is what mathematicians know as a conjugacy class

One starts by finding irreducible representations for the simplest groups first, the Abelian groups, where the number of classes is equal to the order of the group. Then one works up to more complicated groups, utilizing previously determined
representations for the subgroups. This procedure is carried out systematically for many groups in the book "Group Theory" by Hammermesh (Addison-Wesley, 1962), Chapter 4. Several alternative methods are considered. I cannot elaborate more here but hope this might point you in a useful direction. Good luck!